We live in a quantum world, yet, our everyday experiences seem to present themselves in terms of classical information. To examine large quantum systems, we can obtain classical information to tomographically construct a description of the system’s quantum state. However, the number of measurements required to obtain such a description for $N$ qubits scales exponentially with the number of qubits. To avoid this exponential scaling, people have made many efforts to design efficient protocols capable of extracting the relevant properties of a quantum state [1,2,3,4].

In 2020, Huang, Küng, and Preskill proposed a shadow-estimation scheme [5] that can estimate many properties of a quantum state simultaneously using unbiased classical matrix-valued estimators of the state (called classical shadows) generated from randomized measurements (see also [6]). This provides an elegant solution to learning quantum states, as the classical shadows can be stored and reused for different estimation tasks. Thanks to this feature, people can “measure first, ask questions later” [7], making the experiment flexible. Moreover, the recyclability of the classical shadows for learning different properties makes it highly sample-efficient.

However, when it comes to implementation, we need to take many practical device restrictions into account. Previous works have focused on two major practical issues of randomized measurements: noise and feasibility. The noise from random measurements itself can be quite complicated to characterize and mitigate. Meanwhile, the previous shadow estimation scheme based on global random Clifford measurement is challenging to implement with shallow quantum circuits. Owing to the property of randomized measurements, the noise contributions from different random measurements are twirled and mapped to simple models, and can then be effectively mitigated in a robust version of the shadow estimation protocol [8,9]. Additionally, to get rid of the global Clifford measurement, Hu et al. propose a shadow-estimation protocol with shallow local random circuits and construct a recovery map using the measured second Rényi entropy of the input state [10].
They show numerically that the circuit depth can be sublinear in the qubit number of the input state.
Furthermore, Tran et al. recently showed that global random measurements can be realized with a fixed joint Hamiltonian evolution and projective measurements on some ancillary qubits [11], enabling the shadow estimation on an analog quantum simulator.

In recent studies by Zhou and Liu [12] as well as by Helsen and Walter [13], the effect of the switching time between measurement settings on randomized measurements was examined. The original shadow-estimation scheme generated classical-shadow samples from independent and identically distributed settings of randomized measurements, but this “single-shot” randomized measurement is not ideal for many experimental platforms due to the latency involved in circuit compiling and hardware logic. In comparison, the “multi-shot” shadow estimation, where measurements are repeated in a given setting, may be more time-efficient. This has been demonstrated in previous shadow-estimation experiments in optical [14] and superconducting platforms [15] where a shot number of $10^4$ and $10^3$ was chosen, respectively. Recent studies [16,17,18] have also shown that multi-shot random measurements can conserve more samples and measurement settings for the estimation of nonlinear observables, which can be utilized for entanglement quantification.

Zhou and Liu [12] performed a statistical analysis of the multi-shot shadow estimation and showed that the sample complexity is closely related to a new quantity called the cross-shadow norm $\Gamma_2(\rho, O)$ determined by the input state $\rho$ and observable $O$, which characterizes the statistical variance generated by samples with the same measurement setting. For local Pauli measurements and Pauli observables $P$, where $Tr(P\rho)=0$, the estimator variance is proportional to $1/(MK)$, where $M$ and $K$ are the measurement setting and shot number, respectively. In this case, adding the shot number $K$ can suppress the variance as if more measurement settings were added. This property is referred to as shot-efficiency. However, for global Clifford measurements, increasing the number of shots is generally not helpful in suppressing the sample variance. Helsen and Walter [13] proved similar results for global Clifford measurements and showed that introducing unitary $4$-design random unitaries made the shadow-estimation scheme shot-efficient again. However, unitary $4$-design random measurements are not efficient for classical post-processing since they are difficult to simulate classically. As a trade-off between Clifford and unitary $4$-design random measurements, Helsen and Walter studied randomized measurements with a sequence of Clifford gates inserted by $k$ $T$ gates in the middle and provided the optimal shot number $K$ for a given $T$-gate number $k$.

In the future, it would be interesting to study a good randomized measurement schemes that take various practical factors into consideration at the same time. For instance, can global Clifford measurements in multi-shot schemes be replaced with local shallow randomized circuits or Hamiltonian evolution, and can the noise in the shallow measurement circuits be easily mitigated? With smarter protocol designs to bridge the gap between theory and experiment, the shadow estimation and other randomized-measurement protocols will become a powerful toolbox for exploring and understanding the quantum world.

Acknowledgments:
I am grateful to Zhenhuan Liu and Wenjun Yu for helpful discussions during the preparation of this perspective.

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